Norm-resolvent asymptotics for the Neumann Laplacian on periodic resonant thin structures
Abstract
We consider the Neumann Laplacian on a three-dimensional periodic network of thin rods with edge thickness of order h and vertex diameter of order h2/3. We extend the fixed-graph resonant reduction to obtain a finite-h periodic graph approximation with generalised norm-resolvent error O(h), uniformly in the Floquet quasi-momentum. An exact gauge transformation identifies the phase-corrected graph fibres with the canonical lattice Floquet fibres, giving the same rate for the global operators. Collapsing the contact points while retaining the ambient edge tangents gives an approximation with error O(h2/3). The proof uses a Ryzhov boundary triple and a Schur--Feshbach reduction. The estimates yield quantitative approximation of Bloch bands and stability of open gaps on bounded energy intervals. For the simple-cubic lattice, we refine the known resonant dispersion calculation to obtain a complete band-gap decomposition and O(h) control of the persistence of each fixed finite collection of gaps in the thin structure.
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BibTeX
@misc{cherednichenko2026normresolvent,
title = {{Norm-resolvent asymptotics for the Neumann Laplacian on periodic resonant thin structures}},
author = {Kirill Cherednichenko and Chuanjie Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17294},
eprint = {2610.17294},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17294},
}
RIS
TY - UNPB
TI - Norm-resolvent asymptotics for the Neumann Laplacian on periodic resonant thin structures
AU - Kirill Cherednichenko
AU - Chuanjie Wu
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17294
UR - https://arxiv.org/abs/2610.17294
AB - We consider the Neumann Laplacian on a three-dimensional periodic network of thin rods with edge thickness of order h and vertex diameter of order h^2/3. We extend the fixed-graph resonant reduction to obtain a finite-h periodic graph approximation with generalised norm-resolvent error O(h), uniformly in the Floquet quasi-momentum. An exact gauge transformation identifies the phase-corrected graph fibres with the canonical lattice Floquet fibres, giving the same rate for the global operators. Collapsing the contact points while retaining the ambient edge tangents gives an approximation with error O(h^2/3). The proof uses a Ryzhov boundary triple and a Schur--Feshbach reduction. The estimates yield quantitative approximation of Bloch bands and stability of open gaps on bounded energy intervals. For the simple-cubic lattice, we refine the known resonant dispersion calculation to obtain a complete band-gap decomposition and O(h) control of the persistence of each fixed finite collection of gaps in the thin structure.
ER -